Inverse Trigonometric Functions
Trig functions ek angle ko ratio me badalte hain. Kabhi-kabhi ulta chahiye hota hai — ratio se angle nikaalna. Wahi kaam inverse trigonometric functions karte hain, jaise \(\sin^{-1}, \cos^{-1}, \tan^{-1}\).
1. Inverse kyun aur kaise
Trig functions periodic hain, isliye wo one-one nahi — ek ratio kai angles deta hai. Inverse define karne ke liye hum domain ko ek chhote range tak restrict karte hain (principal value branch), taaki function one-one ban jaye.
2. Domain aur Principal Value Range
| Function | Domain | Principal Range |
|---|---|---|
| \(\sin^{-1}x\) | \([-1, 1]\) | \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) |
| \(\cos^{-1}x\) | \([-1, 1]\) | \([0, \pi]\) |
| \(\tan^{-1}x\) | \(\mathbb{R}\) | \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) |
| \(\cot^{-1}x\) | \(\mathbb{R}\) | \((0, \pi)\) |
| \(\sec^{-1}x\) | \(|x| \ge 1\) | \([0, \pi] \setminus \{\frac{\pi}{2}\}\) |
| \(\csc^{-1}x\) | \(|x| \ge 1\) | \([-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}\) |
\(\sin^{-1}\left(\dfrac{1}{2}\right)\) ki principal value. Hum wo angle chahte hain jiska \(\sin\) \(\frac{1}{2}\) ho aur jo \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) me ho: \[ \sin^{-1}\left(\tfrac{1}{2}\right) = \frac{\pi}{6}. \]
3. Basic Properties
\[ \sin^{-1}(\sin x) = x \quad \text{(jab } x \text{ principal range me ho)} \]
\[ \sin(\sin^{-1}x) = x, \qquad x \in [-1, 1] \]
\[ \sin^{-1}(-x) = -\sin^{-1}x, \qquad \tan^{-1}(-x) = -\tan^{-1}x \]
\[ \cos^{-1}(-x) = \pi - \cos^{-1}x \]
4. Complementary Relations
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
\[ \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \]
\[ \sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \]
5. Addition Formula
\[ \tan^{-1}x + \tan^{-1}y = \tan^{-1}\!\left(\frac{x + y}{1 - xy}\right), \qquad xy < 1. \]
\(\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi\) (ek famous result). Pehle do ko jodo: \[ \tan^{-1}1 + \tan^{-1}2 = \pi + \tan^{-1}\!\left(\frac{1+2}{1-2}\right) = \pi - \tan^{-1}3, \] isliye total \(= \pi\).
Key Takeaways
- Inverse trig functions ratio se angle dete hain; domain restrict karke one-one banate hain.
- Principal ranges yaad rakho (table) — answer usi range me hona chahiye.
- \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\) (aur tan/cot, sec/csc ke liye bhi).
- \(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy}\) (\(xy < 1\)).